Mikey - if you download the parts manual for the BS 90 & 100 on the last page (60) it lists what the engine number should start with for each model.If it starts with 20 it would a 100cc model, otherwise 12,14, 17, or 18 is some form of 90cc.Screen shot of page 60 is here...

think that what you have is probably a BS50 Sport. As mentioned by "Old Smokey", a 90 would have ID numbers starting 12, 14, 17 or 18, and a 100 would have an ID number starting with 20. I think what has happened is that you have missed the leading"1" in your frame and steering head numbers. The leading "1" can sometimes be mistaken for a scratch on the plate and it usually comes right under one of the attachment rivets.

If I assume that your bike's number is actually 13T021295, that would put it right in the middle of the known number range for a BS50 Sport built in August 1965.

I hope that solves your problem - and good luck with restoring your bike.